Abstract group 288.960: $D_4:S_3^2$

• Label: 288.960
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $14$
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$D_4:S_3^2$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_6^2:D_4^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	71	8	56	64	88	288
Conjugacy classes	1	9	3	10	11	11	45
Divisions	1	9	3	8	11	9	41
Autjugacy classes	1	5	2	4	5	4	21

Dimension	1	2	4	8
Irr. complex chars.	16	20	8	1	45
Irr. rational chars.	16	16	6	3	41

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	$851760$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{6}=d^{12}=[a,b]=[b,d]=1, c^{a}=cd^{6}, d^{a}=d^{5}, c^{b}=c^{5}d^{6}, d^{c}=d^{7} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,5)(4,6)(9,13)(11,14), (1,2)(3,6)(4,5)(9,13)(11,14), (7,8,10,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 15 & 17 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 26 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 5 & 24 \\ 24 & 5 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Transitive group:	24T608				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_4$ $\,\rtimes\,$ $S_3^2$	$(D_4:S_3)$ $\,\rtimes\,$ $S_3$	$(C_4\times S_3)$ $\,\rtimes\,$ $D_6$	$(C_3:Q_8)$ $\,\rtimes\,$ $D_6$	all 29
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $D_6^2$	$C_6^2$ . $C_2^3$	$C_4$ . $(S_3\times D_6)$	$D_6$ . $(C_2\times D_6)$	all 19

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 1346 subgroups in 355 conjugacy classes, 110 normal (18 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6^2$
Commutator:	$G' \simeq$ $C_3\times C_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_4:S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 288: $D_4:S_3^2$

Order 144: $D_6.D_6$ x 4, $C_{12}.D_6$ x 2, $D_6:D_6$ x 2, $C_2^2.S_3^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$, $C_4\times S_3^2$, $C_{12}.D_6$

Order 96: $D_4:D_6$ x 2

Order 72: $C_6.D_6$ x 7, $C_6\wr C_2$ x 4, $C_6:C_{12}$ x 4, $C_3:D_{12}$ x 4, $C_6:D_6$ x 2, $C_6^2:C_2$ x 2, $S_3\times C_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $C_6.D_6$ x 2, $S_3\times D_6$, $D_4\times C_3^2$, $C_3:D_{12}$, $C_{12}:S_3$

Order 48: $S_3\times D_4$ x 7, $D_4:S_3$ x 6, $D_{12}:C_2$ x 6, $C_4\times D_6$ x 6, $D_4:C_6$ x 2, $D_{12}:C_2$ x 2, $S_3\times Q_8$ x 2

Order 36: $C_6:S_3$ x 7, $C_3:C_{12}$ x 6, $C_6^2$ x 2, $C_6\times S_3$ x 2, $S_3^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $D_4:C_2^2$

Order 24: $C_4\times S_3$ x 21, $C_3:D_4$ x 14, $C_2\times D_6$ x 8, $D_{12}$ x 7, $C_3\times D_4$ x 7, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $C_3:Q_8$ x 6, $C_3\times Q_8$ x 2

Order 18: $C_3:S_3$ x 4, $C_3\times C_6$ x 3, $C_3\times S_3$ x 2

Order 16: $D_4:C_2$ x 8, $C_2\times D_4$ x 3, $C_2^2\times C_4$ x 3, $C_2\times Q_8$

Order 12: $D_6$ x 27, $C_{12}$ x 9, $C_3:C_4$ x 9, $C_2\times C_6$ x 8

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 16, $D_4$ x 12, $Q_8$ x 4, $C_2^3$ x 3

Order 6: $S_3$ x 14, $C_6$ x 11

Order 4: $C_2^2$ x 13, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $D_4:S_3^2$

Order 144: $C_{12}:D_6$, $C_{12}.D_6$, $D_6:D_6$, $C_2^2.S_3^2$, $D_6.D_6$, $C_4\times S_3^2$, $C_{12}.D_6$, $C_{12}.D_6$

Order 96: $D_4:D_6$

Order 72: $C_6.D_6$ x 3, $C_6:D_6$, $S_3\times D_6$, $D_4\times C_3^2$, $C_6^2:C_2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_6\wr C_2$, $C_6:C_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3:D_{12}$, $C_6.D_6$

Order 48: $S_3\times D_4$ x 3, $D_4:S_3$ x 2, $D_{12}:C_2$ x 2, $C_4\times D_6$ x 2, $D_4:C_6$, $D_{12}:C_2$, $S_3\times Q_8$

Order 36: $C_6:S_3$ x 3, $C_3:C_{12}$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6^2$, $C_6\times S_3$, $S_3^2$

Order 32: $D_4:C_2^2$

Order 24: $C_4\times S_3$ x 8, $C_3:D_4$ x 4, $D_{12}$ x 3, $C_2\times D_6$ x 3, $C_3\times D_4$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_3:Q_8$ x 2, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 2, $C_3:S_3$ x 2, $C_3\times S_3$

Order 16: $D_4:C_2$ x 3, $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $C_2\times Q_8$

Order 12: $D_6$ x 8, $C_{12}$ x 4, $C_3:C_4$ x 4, $C_2\times C_6$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 7, $D_4$ x 5, $C_2^3$ x 2, $Q_8$ x 2

Order 6: $C_6$ x 5, $S_3$ x 5

Order 4: $C_2^2$ x 6, $C_4$ x 4

Order 3: $C_3$ x 2

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $D_4:S_3^2$ ($C_1$)

Order 144: $D_6.D_6$ ($C_2$) x 4, $C_{12}.D_6$ ($C_2$) x 2, $D_6:D_6$ ($C_2$) x 2, $C_2^2.S_3^2$ ($C_2$) x 2, $C_{12}.D_6$ ($C_2$) x 2, $C_{12}:D_6$ ($C_2$), $C_4\times S_3^2$ ($C_2$), $C_{12}.D_6$ ($C_2$)

Order 72: $C_6.D_6$ ($C_2^2$) x 7, $C_6\wr C_2$ ($C_2^2$) x 4, $C_6:C_{12}$ ($C_2^2$) x 4, $C_3:D_{12}$ ($C_2^2$) x 4, $C_6:D_6$ ($C_2^2$) x 2, $C_6^2:C_2$ ($C_2^2$) x 2, $S_3\times C_{12}$ ($C_2^2$) x 2, $C_3^2:Q_8$ ($C_2^2$) x 2, $C_3^2:Q_8$ ($C_2^2$) x 2, $C_6.D_6$ ($C_2^2$) x 2, $S_3\times D_6$ ($C_2^2$), $D_4\times C_3^2$ ($C_2^2$), $C_3:D_{12}$ ($C_2^2$), $C_{12}:S_3$ ($C_2^2$)

Order 48: $D_4:S_3$ ($S_3$) x 2

Order 36: $C_3:C_{12}$ ($C_2^3$) x 6, $C_6:S_3$ ($C_2^3$) x 3, $C_6^2$ ($C_2^3$) x 2, $C_6\times S_3$ ($C_2^3$) x 2, $C_3\times C_{12}$ ($C_2^3$), $C_3^2:C_4$ ($C_2^3$)

Order 24: $C_3:D_4$ ($D_6$) x 4, $C_6:C_4$ ($D_6$) x 4, $C_4\times S_3$ ($D_6$) x 2, $C_3:Q_8$ ($D_6$) x 2, $C_3\times D_4$ ($D_6$) x 2

Order 18: $C_3:S_3$ ($D_4:C_2$) x 2, $C_3\times C_6$ ($C_2^4$)

Order 12: $C_3:C_4$ ($C_2\times D_6$) x 6, $C_2\times C_6$ ($C_2\times D_6$) x 4, $D_6$ ($C_2\times D_6$) x 2, $C_{12}$ ($C_2\times D_6$) x 2

Order 9: $C_3^2$ ($D_4:C_2^2$)

Order 8: $D_4$ ($S_3^2$)

Order 6: $C_6$ ($C_2^2\times D_6$) x 2

Order 4: $C_2^2$ ($S_3\times D_6$) x 2, $C_4$ ($S_3\times D_6$)

Order 3: $C_3$ ($D_4:D_6$) x 2

Order 2: $C_2$ ($D_6^2$)

Order 1: $C_1$ ($D_4:S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $D_4:S_3^2$ ($C_1$)

Order 144: $C_{12}:D_6$ ($C_2$), $C_{12}.D_6$ ($C_2$), $D_6:D_6$ ($C_2$), $C_2^2.S_3^2$ ($C_2$), $D_6.D_6$ ($C_2$), $C_4\times S_3^2$ ($C_2$), $C_{12}.D_6$ ($C_2$), $C_{12}.D_6$ ($C_2$)

Order 72: $C_6.D_6$ ($C_2^2$) x 3, $C_6:D_6$ ($C_2^2$), $S_3\times D_6$ ($C_2^2$), $D_4\times C_3^2$ ($C_2^2$), $C_6^2:C_2$ ($C_2^2$), $C_3:D_{12}$ ($C_2^2$), $C_{12}:S_3$ ($C_2^2$), $C_6\wr C_2$ ($C_2^2$), $C_6:C_{12}$ ($C_2^2$), $S_3\times C_{12}$ ($C_2^2$), $C_3^2:Q_8$ ($C_2^2$), $C_3^2:Q_8$ ($C_2^2$), $C_3:D_{12}$ ($C_2^2$), $C_6.D_6$ ($C_2^2$)

Order 48: $D_4:S_3$ ($S_3$)

Order 36: $C_3:C_{12}$ ($C_2^3$) x 2, $C_6:S_3$ ($C_2^3$) x 2, $C_3\times C_{12}$ ($C_2^3$), $C_3^2:C_4$ ($C_2^3$), $C_6^2$ ($C_2^3$), $C_6\times S_3$ ($C_2^3$)

Order 24: $C_3:D_4$ ($D_6$), $C_6:C_4$ ($D_6$), $C_4\times S_3$ ($D_6$), $C_3:Q_8$ ($D_6$), $C_3\times D_4$ ($D_6$)

Order 18: $C_3\times C_6$ ($C_2^4$), $C_3:S_3$ ($D_4:C_2$)

Order 12: $C_3:C_4$ ($C_2\times D_6$) x 2, $C_2\times C_6$ ($C_2\times D_6$), $D_6$ ($C_2\times D_6$), $C_{12}$ ($C_2\times D_6$)

Order 9: $C_3^2$ ($D_4:C_2^2$)

Order 8: $D_4$ ($S_3^2$)

Order 6: $C_6$ ($C_2^2\times D_6$)

Order 4: $C_2^2$ ($S_3\times D_6$), $C_4$ ($S_3\times D_6$)

Order 3: $C_3$ ($D_4:D_6$)

Order 2: $C_2$ ($D_6^2$)

Order 1: $C_1$ ($D_4:S_3^2$)

Series

Derived series

$D_4:S_3^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$D_4:S_3^2$

$\rhd$

$C_4\times S_3^2$

$\rhd$

$S_3\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_4:S_3^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 39 larger groups in the database.

This group is a maximal quotient of 187 larger groups in the database.

Character theory

Complex character table

See the $45 \times 45$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $41 \times 41$ rational character table.